A polynomially bounded operator on Hilbert space which is not similar to a contraction

Abstract

Let $\eps >0$. We prove that there exists an operator $T_\eps:\ell_2\to\ell_2$, such that for any polynomial $P$ we have $\|{P(T)}\| \leq(1+\eps)\|{P}\|_\infty$, but which is not similar to a contraction, {\it i.e.} there does not exist an invertible operator $S:{\ell }_2\to {\ell }_2$ such that $\|{S^{-1}T_\eps S}\|\leq 1$. This answers negatively a question attributed to Halmos after his well known 1970 paper (``Ten problems in Hilbert space").

Cited by in corpus (6)

• The norm and modulus of a Foguel operator
• Weak product spaces of Dirichlet series
• Some sufficient conditions for existence of hyperinvariant subspaces for operators intertwined with unitaries
• On existence of shift-type invariant subspaces for polynomially bounded operator
• Vectorial Hankel operators, Carleson embeddings, and notions of $BMOA$
• Invertibility of functions of operators and existence of hyperinvariant subspaces